This invention relates to reciprocating saws, and to abrasive sanding attachments and blades for such saws, to facilitate the oscillating, the reciprocating sanding, and the shaping of straight, curved, concave or convex surfaces, with abrasive cutting surfaces. The invention described herein fits and mounts in blade holders currently in use on scroll saws, and is designed to take advantage of pin-end and plain pinless-end saw blade mounting methods as applied to holding an abrasive member. This invention increases the versatility and use of the scroll saw and related C-shaped saws, as tools for use in intricate, delicate, and controlled woodworking, metalworking, and finishing. Methods for making abrasive member holders 40, 68, 74, 84, 104, 112, 124, 142, and 160 are disclosed. The holders can be mounted on pin-end blade or pin-less blade scroll saws using abrasive member head, extenders and pivot heads.